Salt Lake City previously hosted the Winter Olympics in 2002 but the Games were marred by allegations of bribery between the successful bidders and members of the International Olympic Committee (IOC).
"Salt Lake City and the great state of Utah are ready, willing and able to host a future Olympic Games," Herbert told reporters during a news conference.
"Utah has become the winter sports capital of the USA. By hosting another Winter Games, our commitment to support ongoing Olympic movement initiatives will only be strengthened."
Several other American cities have also expressed interest in hosting the Games but it is up to the United States Olympic Committee (USOC) to decide whether to proceed with a bid.
"At this point we are focused on moving forward with our bid working group that is due to make its first report to the USOC board of directors on December 20th," a USOC spokesman said.
"We are obviously appreciative that the Olympic movement is strong enough in the U.S. to have cities interested in potentially bidding, however the USOC is focused on determining when it would be the right time to potentially bid for an Olympic Games."
The IOC is not due to choose the host site for 2026 until 2019 after already awarding 2014 to Sochi, Russia, and 2018 to Pyeonchang, South Korea.
The 2022 host will be decided in 2015. The US has already ruled out making a bid for 2022.